LAPACK  3.10.1 LAPACK: Linear Algebra PACKage
sort01.f
Go to the documentation of this file.
1 *> \brief \b SORT01
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 * Definition:
9 * ===========
10 *
11 * SUBROUTINE SORT01( ROWCOL, M, N, U, LDU, WORK, LWORK, RESID )
12 *
13 * .. Scalar Arguments ..
14 * CHARACTER ROWCOL
15 * INTEGER LDU, LWORK, M, N
16 * REAL RESID
17 * ..
18 * .. Array Arguments ..
19 * REAL U( LDU, * ), WORK( * )
20 * ..
21 *
22 *
23 *> \par Purpose:
24 * =============
25 *>
26 *> \verbatim
27 *>
28 *> SORT01 checks that the matrix U is orthogonal by computing the ratio
29 *>
30 *> RESID = norm( I - U*U' ) / ( n * EPS ), if ROWCOL = 'R',
31 *> or
32 *> RESID = norm( I - U'*U ) / ( m * EPS ), if ROWCOL = 'C'.
33 *>
34 *> Alternatively, if there isn't sufficient workspace to form
35 *> I - U*U' or I - U'*U, the ratio is computed as
36 *>
37 *> RESID = abs( I - U*U' ) / ( n * EPS ), if ROWCOL = 'R',
38 *> or
39 *> RESID = abs( I - U'*U ) / ( m * EPS ), if ROWCOL = 'C'.
40 *>
41 *> where EPS is the machine precision. ROWCOL is used only if m = n;
42 *> if m > n, ROWCOL is assumed to be 'C', and if m < n, ROWCOL is
43 *> assumed to be 'R'.
44 *> \endverbatim
45 *
46 * Arguments:
47 * ==========
48 *
49 *> \param[in] ROWCOL
50 *> \verbatim
51 *> ROWCOL is CHARACTER
52 *> Specifies whether the rows or columns of U should be checked
53 *> for orthogonality. Used only if M = N.
54 *> = 'R': Check for orthogonal rows of U
55 *> = 'C': Check for orthogonal columns of U
56 *> \endverbatim
57 *>
58 *> \param[in] M
59 *> \verbatim
60 *> M is INTEGER
61 *> The number of rows of the matrix U.
62 *> \endverbatim
63 *>
64 *> \param[in] N
65 *> \verbatim
66 *> N is INTEGER
67 *> The number of columns of the matrix U.
68 *> \endverbatim
69 *>
70 *> \param[in] U
71 *> \verbatim
72 *> U is REAL array, dimension (LDU,N)
73 *> The orthogonal matrix U. U is checked for orthogonal columns
74 *> if m > n or if m = n and ROWCOL = 'C'. U is checked for
75 *> orthogonal rows if m < n or if m = n and ROWCOL = 'R'.
76 *> \endverbatim
77 *>
78 *> \param[in] LDU
79 *> \verbatim
80 *> LDU is INTEGER
81 *> The leading dimension of the array U. LDU >= max(1,M).
82 *> \endverbatim
83 *>
84 *> \param[out] WORK
85 *> \verbatim
86 *> WORK is REAL array, dimension (LWORK)
87 *> \endverbatim
88 *>
89 *> \param[in] LWORK
90 *> \verbatim
91 *> LWORK is INTEGER
92 *> The length of the array WORK. For best performance, LWORK
93 *> should be at least N*(N+1) if ROWCOL = 'C' or M*(M+1) if
94 *> ROWCOL = 'R', but the test will be done even if LWORK is 0.
95 *> \endverbatim
96 *>
97 *> \param[out] RESID
98 *> \verbatim
99 *> RESID is REAL
100 *> RESID = norm( I - U * U' ) / ( n * EPS ), if ROWCOL = 'R', or
101 *> RESID = norm( I - U' * U ) / ( m * EPS ), if ROWCOL = 'C'.
102 *> \endverbatim
103 *
104 * Authors:
105 * ========
106 *
107 *> \author Univ. of Tennessee
108 *> \author Univ. of California Berkeley
109 *> \author Univ. of Colorado Denver
110 *> \author NAG Ltd.
111 *
112 *> \ingroup single_eig
113 *
114 * =====================================================================
115  SUBROUTINE sort01( ROWCOL, M, N, U, LDU, WORK, LWORK, RESID )
116 *
117 * -- LAPACK test routine --
118 * -- LAPACK is a software package provided by Univ. of Tennessee, --
119 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
120 *
121 * .. Scalar Arguments ..
122  CHARACTER ROWCOL
123  INTEGER LDU, LWORK, M, N
124  REAL RESID
125 * ..
126 * .. Array Arguments ..
127  REAL U( LDU, * ), WORK( * )
128 * ..
129 *
130 * =====================================================================
131 *
132 * .. Parameters ..
133  REAL ZERO, ONE
134  parameter( zero = 0.0e+0, one = 1.0e+0 )
135 * ..
136 * .. Local Scalars ..
137  CHARACTER TRANSU
138  INTEGER I, J, K, LDWORK, MNMIN
139  REAL EPS, TMP
140 * ..
141 * .. External Functions ..
142  LOGICAL LSAME
143  REAL SDOT, SLAMCH, SLANSY
144  EXTERNAL lsame, sdot, slamch, slansy
145 * ..
146 * .. External Subroutines ..
147  EXTERNAL slaset, ssyrk
148 * ..
149 * .. Intrinsic Functions ..
150  INTRINSIC max, min, real
151 * ..
152 * .. Executable Statements ..
153 *
154  resid = zero
155 *
156 * Quick return if possible
157 *
158  IF( m.LE.0 .OR. n.LE.0 )
159  \$ RETURN
160 *
161  eps = slamch( 'Precision' )
162  IF( m.LT.n .OR. ( m.EQ.n .AND. lsame( rowcol, 'R' ) ) ) THEN
163  transu = 'N'
164  k = n
165  ELSE
166  transu = 'T'
167  k = m
168  END IF
169  mnmin = min( m, n )
170 *
171  IF( ( mnmin+1 )*mnmin.LE.lwork ) THEN
172  ldwork = mnmin
173  ELSE
174  ldwork = 0
175  END IF
176  IF( ldwork.GT.0 ) THEN
177 *
178 * Compute I - U*U' or I - U'*U.
179 *
180  CALL slaset( 'Upper', mnmin, mnmin, zero, one, work, ldwork )
181  CALL ssyrk( 'Upper', transu, mnmin, k, -one, u, ldu, one, work,
182  \$ ldwork )
183 *
184 * Compute norm( I - U*U' ) / ( K * EPS ) .
185 *
186  resid = slansy( '1', 'Upper', mnmin, work, ldwork,
187  \$ work( ldwork*mnmin+1 ) )
188  resid = ( resid / real( k ) ) / eps
189  ELSE IF( transu.EQ.'T' ) THEN
190 *
191 * Find the maximum element in abs( I - U'*U ) / ( m * EPS )
192 *
193  DO 20 j = 1, n
194  DO 10 i = 1, j
195  IF( i.NE.j ) THEN
196  tmp = zero
197  ELSE
198  tmp = one
199  END IF
200  tmp = tmp - sdot( m, u( 1, i ), 1, u( 1, j ), 1 )
201  resid = max( resid, abs( tmp ) )
202  10 CONTINUE
203  20 CONTINUE
204  resid = ( resid / real( m ) ) / eps
205  ELSE
206 *
207 * Find the maximum element in abs( I - U*U' ) / ( n * EPS )
208 *
209  DO 40 j = 1, m
210  DO 30 i = 1, j
211  IF( i.NE.j ) THEN
212  tmp = zero
213  ELSE
214  tmp = one
215  END IF
216  tmp = tmp - sdot( n, u( j, 1 ), ldu, u( i, 1 ), ldu )
217  resid = max( resid, abs( tmp ) )
218  30 CONTINUE
219  40 CONTINUE
220  resid = ( resid / real( n ) ) / eps
221  END IF
222  RETURN
223 *
224 * End of SORT01
225 *
226  END
subroutine slaset(UPLO, M, N, ALPHA, BETA, A, LDA)
SLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition: slaset.f:110
subroutine ssyrk(UPLO, TRANS, N, K, ALPHA, A, LDA, BETA, C, LDC)
SSYRK
Definition: ssyrk.f:169
subroutine sort01(ROWCOL, M, N, U, LDU, WORK, LWORK, RESID)
SORT01
Definition: sort01.f:116